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Chapter 39

Relativity

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A Brief Overview of Modern

Physics 20th Century revolution

1900 Max Planck Basic ideas leading to Quantum theory

1905 Einstein Special Theory of Relativity

21st Century Story is still incomplete

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Basic Problems Newtonian mechanics fails to describe

properly the motion of objects whose

speeds approach that of light Newtonian mechanics is a limited theory

It places no upper limit on speed

It is contrary to modern experimental results Newtonian mechanics becomes a specialized

case of Einsteins special theory of relativity When speeds are much less than the speed of light

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Galilean Relativity To describe a physical event, a frame of

reference must be established

There is no absolute inertial frame ofreference This means that the results of an

experiment performed in a vehicle movingwith uniform velocity will be identical to theresults of the same experiment performedin a stationary vehicle

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Galilean Relativity, cont. Reminders about inertial frames

Objects subjected to no forces will experience no

acceleration Any system moving at constant velocity with

respect to an inertial frame must also be in an

inertial frame

According to the principle of Galileanrelativity, the laws of mechanics are the

same in all inertial frames of reference

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Galilean Relativity Example The observer in the

truck throws a ball

straight up It appears to move in

a vertical path

The law of gravity

and equations ofmotion under uniform

acceleration are

obeyed

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Galilean Relativity Example,

cont.

There is a stationary observer on the ground Views the path of the ball thrown to be a parabola The ball has a velocity to the right equal to the

velocity of the truck

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Galilean Relativity Example,

conclusion The two observers disagree on the shape of

the balls path

Both agree that the motion obeys the law ofgravity and Newtons laws of motion

Both agree on how long the ball was in the air

Conclusion: There is no preferred frame ofreference for describing the laws of

mechanics

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Views of an Event An eventis some

physical phenomenon

Assume the eventoccurs and is observedby an observer at rest inan inertial reference

frame The events location

and time can bespecified by the

coordinates (x, y, z, t)

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Views of an Event, cont. Consider two inertial frames, S and S

S moves with constant velocity, v,

along the commonxandx axes

The velocity is measured relative to S

Assume the origins of S and S coincide

at t= 0

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Galilean Space-Time

Transformation Equations An observer in S describes the event with

space-time coordinates (x, y, z, t)

An observer in S describes the same eventwith space-time coordinates (x, y, z, t)

The relationship among the coordinates are x =x vt

y = y

z = z

t = t

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Notes About Galilean

Transformation Equations The time is the same in both inertial

frames

Within the framework of classicalmechanics, all clocks run at the same rate

The time at which an event occurs for anobserver in S is the same as the time for

the same event in S This turns out to be incorrect when vis

comparable to the speed of light

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Speed of Light Galilean relativity does not apply to electricity,

magnetism, or optics

Maxwell showed the speed of light in freespace is c= 3.00 x 108 m/s

Physicists in the late 1800s thought light

moved through a medium called the ether The speed of light would be conly in a special,

absolute frame at rest with respect to the ether

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Effect of Ether Wind

on Light Assume vis the

velocity of the ether

wind relative to theearth

cis the speed of

light relative to the

ether Various resultant

velocities are shown

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Ether Wind, cont. The velocity of the ether wind is assumed to

be the orbital velocity of the Earth

All attempts to detect and establish theexistence of the ether wind proved futile

But Maxwells equations seem to imply that

the speed of light always has a fixed value in

all inertial frames This is a contradiction to what is expected based

on the Galilean velocity transformation equation

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Michelson-Morley Experiment First performed in 1881 by Michelson

Repeated under various conditions by

Michelson and Morley

Designed to detect small changes in the

speed of light By determining the velocity of the Earth

relative to the ether

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Michelson-Morley Equipment Used the Michelson

interferometer Arm 2 is aligned along the

direction of the Earths motionthrough space

The interference pattern wasobserved while theinterferometer was rotated

through 90 The effect should have been

to show small, butmeasurable, shifts in thefringe pattern

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Active Figure 39.4

(SLIDESHOW MODE ONLY)

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Michelson-Morley Results Measurements failed to show any change in

the fringe pattern No fringe shift of the magnitude required was ever

observed The negative results contradicted the ether hypothesis They also showed that it was impossible to measure

the absolute velocity of the Earth with respect to theether frame

Light is now understood to be anelectromagnetic wave, which requires nomedium for its propagation

The idea of an ether was discarded

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Albert Einstein 1879 1955

1905 Special theory of relativity

1916 General relativity

1919 confirmation

1920s Didnt accept quantumtheory

1940s or so Search for unified theory -

unsuccessful

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Einsteins Principle of

Relativity Resolves the contradiction between Galilean

relativity and the fact that the speed of light isthe same for all observers

Postulates The principle of relativity: The laws of physics

must be the same in all inertial reference frames The constancy of the speed of light: the speed

of light in a vacuum has the same value, c= 3.00x 108 m/s, in all inertial reference frames,regardless of the velocity of the observer or thevelocity of the source emitting the light

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The Constancy of the Speed

of Light This is required by the first postulate Confirmed experimentally in many ways Explains the null result of the

Michelson-Morley experiment Relative motion is unimportant when

measuring the speed of light We must alter our common-sense notions

of space and time

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Consequences of Special

Relativity Restricting the discussion to concepts of

simultaneity, time intervals, and length These are quite different in relativistic mechanics

from what they are in Newtonian mechanics

In relativistic mechanics There is no such thing as absolute length There is no such thing as absolute time Events at different locations that are observed to

occur simultaneously in one frame are notobserved to be simultaneous in another framemoving uniformly past the first

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Simultaneity In special relativity, Einstein abandoned the

assumption of simultaneity

Thought experiment to show this A boxcar moves with uniform velocity

Two lightning bolts strike the ends

The lightning bolts leave marks (A and B) on the

car and (A and B) on the ground Two observers are present: O in the boxcar and O

on the ground

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Simultaneity Thought

Experiment Set-up

ObserverO is midway between the points of

lightning strikes on the ground,A and B

ObserverO is midway between the points of

lightning strikes on the boxcar,A and B

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Simultaneity Thought

Experiment Results

The light reaches observerO at the same time He concludes the light has traveled at the same

speed over equal distances

ObserverO concludes the lightning bolts occurred

simultaneously

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Simultaneity Thought

Experiment Results, cont. By the time the light has

reached observerO, observerO has moved

The signal from B hasalready swept past O, but thesignal fromA has not yetreached him The two observers must find

that light travels at the samespeed

ObserverO concludes thelightning struck the front ofthe boxcar before it struckthe back (they were not

simultaneous events)

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Simultaneity Thought

Experiment, Summary Two events that are simultaneous in one

reference frame are in general notsimultaneous in a second reference framemoving relative to the first

That is, simultaneity is not an absoluteconcept, but rather one that depends on thestate of motion of the observer In the thought experiment, both observers are

correct, because there is no preferred inertialreference frame

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Simultaneity, Transit Time In this thought experiment, the

disagreement depended upon the

transit time of light to the observers anddoesnt demonstrate the deepermeaning of relativity

In high-speed situations, thesimultaneity is relative even whentransit time is subtracted out We will ignore transit time in all further discussions

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Time Dilation A mirror is fixed to the ceiling

of a vehicle The vehicle is moving to the

right with speed v An observer, O, at rest in the

frame attached to the vehicleholds a flashlight a distance dbelow the mirror

The flashlight emits a pulse oflight directed at the mirror(event 1) and the pulse arrivesback after being reflected(event 2)

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Time Dilation, Moving

Observer ObserverO carries a clock

She uses it to measure the time

between the events (tp)

She observes the events to occur at the

same place

tp = distance/speed = (2d)/c

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Time Dilation, Observations Both observers must measure the

speed of the light to be c

The light travels farther forO The time interval, t, forO is longer

than the time interval forO, tp

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Time Dilation, Time

Comparisons

2

2

2

2

1

1where

1

p

p

tt t

v

c

vc

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Time Dilation, Summary The time interval tbetween two events

measured by an observer moving with

respect to a clock is longer than thetime interval tp between the same two

events measured by an observer at rest

with respect to the clock This is known as time dilation

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Active Figure 39.6

(SLIDESHOW MODE ONLY)

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Factor Table

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Identifying Proper Time The time interval tp is called the

proper time interval The proper time interval is the time interval

between events as measured by an

observer who sees the events occur at the

same point in space You must be able to correctly identify the

observer who measures the proper time

interval

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Time Dilation Generalization If a clock is moving with respect to you, the

time interval between ticks of the movingclock is observed to be longer that the timeinterval between ticks of an identical clock inyour reference frame

All physical processes are measured to slowdown when these processes occur in a framemoving with respect to the observer These processes can be chemical and biological

as well as physical

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Time Dilation Verification Time dilation is a very real phenomenon

that has been verified by various

experiments These experiments include:

Airplane flights

Muon decay Twin Paradox

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Time Dilation Verification

Muon Decays Muons are unstable particles that

have the same charge as an electron,

but a mass 207 times more than an

electron Muons have a half-life of tp = 2.2s

when measured in a reference frame

at rest with respect to them (a)

Relative to an observer on the Earth,

muons should have a lifetime of

tp (b)

A CERN experiment measured

lifetimes in agreement with the

predictions of relativity

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The Twin Paradox The

Situation A thought experiment involving a set of twins,

Speedo and Goslo

Speedo travels to Planet X, 20 light yearsfrom the Earth His ship travels at 0.95c

After reaching Planet X, he immediately returns to

the Earth at the same speed When Speedo returns, he has aged 13 years,

but Goslo has aged 42 years

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The Twins Perspectives Goslos perspective is that he was at

rest while Speedo went on the journey

Speedo thinks he was at rest and Gosloand the Earth raced away from him and

then headed back toward him

The paradox which twin hasdeveloped signs of excess aging?

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The Twin Paradox The

Resolution Relativity applies to reference frames moving

at uniform speeds

The trip in this thought experiment is notsymmetrical since Speedo must experience aseries of accelerations during the journey

Therefore, Goslo can apply the time dilation

formula with a proper time of 42 years This gives a time for Speedo of 13 years and thisagrees with the earlier result

There is no true paradox since Speedo is not

in an inertial frame

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Length Contraction Equation

Length contraction

takes place only

along the direction

of motion

2

21P P

L vL L

c

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Active Figure 39.11

(SLIDESHOW MODE ONLY)

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Length Contraction, Final The observer who measures the proper

length must be correctly identified

The proper length between two points inspace is always the length measured by

an observer at rest with respect to the

points

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Proper Length vs. Proper

Time The proper length and proper time

interval are defined differently

The proper length is measured by anobserver for whom the end points of thelength remained fixed in space

The proper time interval is measured bysomeone for whom the two events takeplace at the same position in space

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Space-Time Graphs In a space-time graph,

ctis the ordinate and

positionxis the abscissa

The example is the graph

of the twin paradox

A path through space-time

is called a world-line

World-lines for light are

diagonal lines

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Relativistic Doppler Effect Another consequence of time dilation is

the shift in frequency found for light

emitted by atoms in motion as opposed tolight emitted by atoms at rest

If a light source and an observer approach

each other with a relative speed, v, the

frequency measured by the observer is

obs source

1

1

v c

v c

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Relativistic Doppler Effect,

cont. The frequency of the source is

measured in its rest frame

The shift depends only on the relativevelocity, v, of the source and observer

obs > source when the source and the

observer approach each other An example is the red shiftof galaxies,

showing most galaxies are movingaway from us

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Lorentz Transformation

Equations, Set-Up, cont.

The observer in frame S reports the eventwith space-time coordinates of (x, y, z, t)

The observer in frame S reports the sameevent with space-time coordinates of (x, y,z, t)

If two events occur, at points P and Q, then

the Galilean transformation would predict thatx= x The distance between the two points in space at

which the events occur does not depend on themotion of the observer

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Lorentz Transformations,

Equations

To transform coordinates from S to S use

These show that in relativity, space and time

are not separate concepts but rather closely

interwoven with each other

To transform coordinates from S to S use

2

' ' ' ' v

x x vt y y z z t t x

c

2

' ' ' ' ' ' v

x x vt y y z z t t x c

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Lorentz Transformations,

Pairs of Events

The Lorentz transformations can be

written in a form suitable for describing

pairs of events For S to S For S to S

2

'

'

x x v t

vt t x

c

2

' '

' '

x x v t

vt t x c

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L V l i

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Lorentz Velocity

Transformation, cont.

The term vdoes not appear in the uy and uz

equations since the relative motion is in thex

direction When vis much smaller than c, the Lorentz

velocity transformation reduces to the

Galilean velocity transformation equation

Ifv= c, ux = cand the speed of light is

shown to be independent of the relative

motion of the frame

L t V l it

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Lorentz Velocity

Transformation, final

To obtain ux in terms ofux, use

21

'

'

x

xx

u v

u u v

c

M t Ob D

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Measurements Observers Do

Not Agree On

Two observers O and O do not agreeon:

The time interval between events that takeplace in the same position in one referenceframe

The distance between two points that remainfixed in one of their frames

The velocity components of a moving particle Whether two events occurring at different

locations in both frames are simultaneous ornot

M t Ob D

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Measurements Observers Do

Agree On

Two observers O and O can agree on: Their relative speed of motion vwith

respect to each other The speed cof any ray of light

The simultaneity of two events which take

place at the same position andtime insome frame

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Relativistic Linear Momentum

To account for conservation of momentum inall inertial frames, the definition must bemodified to satisfy these conditions The linear momentum of an isolated particle must be

conserved in all collisions The relativistic value calculated for the linear momentum

p of a particle must approach the classical value mu asu approaches zero

u is the velocity of the particle, m is its mass

2

21

m mu

c

up u

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Relativistic Energy

The definition of kinetic energy requires

modification in relativistic mechanics

E = mc2

mc2

This matches the classical kinetic energy equation

when u

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Relativistic Energy

Consequences

A particle has energy by virtue of its

mass alone

A stationary particle with zero kineticenergy has an energy proportional to its

inertial mass

This is shown by E= K+ mc2

A small mass corresponds to an

enormous amount of energy

E d R l ti i ti

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Energy and Relativistic

Momentum

It is useful to have an expressionrelating total energy, E, to the relativistic

momentum, p E2 =p2c2+ (mc

2)2

When the particle is at rest,p = 0 and E= mc2 Massless particles (m = 0) have E=pc

The mass m of a particle is independent ofits motion and so is the same value in allreference frames m is often called the invariant mass

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Mass and Energy

This is also used to express masses inenergy units mass of an electron = 9.11 x 10-31 kg = 0.511 Me Conversion: 1 u = 929.494 MeV/c2

When using Conservation of Energy, restenergy must be included as another form ofenergy storage

The conversion from mass to energy is usefulin nuclear reactions

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More About Mass

Mass has two seemingly different properties A gravitational attraction for other masses: Fg =

mgg

An inertialproperty that represents a resistance toacceleration: F= mia

That mg and mi were directly proportional was

evidence for a connection between them Einsteins view was that the dual behavior of

mass was evidence for a very intimate andbasic connection between the two behaviors

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Elevator Example, 1

The observer is at rest

in a uniform

gravitational field, g,

directed downward He is standing in an

elevator on the surface

of a planet

He feels pressed intothe floor, due to the

gravitational force

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Elevator Example, 2

Here the observer isin a region wheregravity is negligible

A force is producingan upwardacceleration ofa = g

The person feels

pressed to the floorwith the same forceas in the gravitationalfield

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Elevator Example, 3

In both cases, an

object released by

the observer

undergoes a

downward

acceleration ofg

relative to the floor

Elevator Example

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Elevator Example,

Conclusions

Einstein claimed that the two situations

were equivalent

No local experiment can distinguishbetween the two frames One frame is an inertial frame in a

gravitational field The other frame is accelerating in a

gravity-free space

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Einsteins Conclusions, cont.

Einstein extended the idea further and

proposed that no experiment, mechanical or

otherwise, could distinguish between the two

cases

He proposed that a beam of light should be

bent downward by a gravitational field

The bending would be small A laser would fall less than 1 cm from the

horizontal after traveling 6000 km

Postulates of General

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Postulates of General

Relativity

All the laws of nature have the same

form for observers in any frame of

reference, whether accelerated or not In the vicinity of any point, a

gravitational field is equivalent to an

accelerated frame of reference in theabsence of gravitational effects This is the principle of equivalence

Implications of General

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Implications of General

Relativity

Time is altered by gravity A clock in the presence of gravity runs

slower than one where gravity is negligible The frequencies of radiation emitted by

atoms in a strong gravitational field are

shifted to lower frequencies This has been detected in the spectral

lines emitted by atoms in massive stars

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Curvature of Space-Time

The curvature of space-time completely

replaces Newtons gravitational theory

There is no such thing as a gravitational field according to Einstein

Instead, the presence of a mass causes a

curvature of time-space in the vicinity of the

mass This curvature dictates the path that all freely

moving objects must follow

Effect of Curvature of Space

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Effect of Curvature of Space-

Time

Imagine two travelers moving on parallelpaths a few meters apart on the surface ofthe Earth, heading exactly northward

As they approach the North Pole, their pathswill be converging

They will have moved toward each other as ifthere were an attractive force between them

It is the geometry of the curved surface thatcauses them to converge, rather than anattractive force between them

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Testing General Relativity

General relativity predicts that a light ray passingnear the Sun should be deflected in the curvedspace-time created by the Suns mass

The prediction was confirmed by astronomers duringa total solar eclipse

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Einsteins Cross

The four spots areimages of the samegalaxy

They have beenbent around amassive objectlocated between the

galaxy and the Earth The massive object

acts like a lens

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Black Holes

If the concentration of mass becomes

very great, a black hole may form

In a black hole, the curvature of space-time is so great that, within a certain

distance from its center, all light and

matter become trapped